Rules of thumb for divergence/convergence comparisons?

Hey, I was wondering a little about determining convergence/divergence of improper integrals. I have no problems doing the calculations and such, but I still feel a little bit like I'm floundering in the dark when it comes to deciding exactly what function to choose for comparison integrals. For example, if we have the integral from 2 to infinity of

x*sqrt(x)/x^2-1

... I just happen to know that 1/sqrt(x) is suitable for comparison, but I'm not sure I could just guess or conclude that from something specific. So basically I was just wondering if there are any good rules of thumb for this, when it comes to somewhat more complicated functions, like the one above and worse? Or is it really just hitting your head against it until you work up plenty of experience at it?

Re: Rules of thumb for divergence/convergence comparisons?

Quote:

Originally Posted by

**Scurmicurv** Hey, I was wondering a little about determining convergence/divergence of improper integrals. I have no problems doing the calculations and such, but I still feel a little bit like I'm floundering in the dark when it comes to deciding exactly what function to choose for comparison integrals. For example, if we have the integral from 2 to infinity of

x*sqrt(x)/x^2-1

... I just happen to know that 1/sqrt(x) is suitable for comparison, but I'm not sure I could just guess or conclude that from something specific. So basically I was just wondering if there are any good rules of thumb for this, when it comes to somewhat more complicated functions, like the one above and worse? Or is it really just hitting your head against it until you work up plenty of experience at it?

note the degree of the numerator is 3/2 and the degree of the denominator is 2 ...

equivalently, you're looking at an integral of the same degree as $\displaystyle \frac{1}{x^{1/2}}$

Re: Rules of thumb for divergence/convergence comparisons?

Heeey, that... ha, wow, I can't believe I haven't seen that myself before. But yea, I went back and looked over a few old assignments and had a fair amount of stuff clicking into place. Thanks a lot for pointing that out!